1985 Putnam B3

Let

$$

\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\

a_{2,1} & a_{2,2} & a_{2,3} & \dots \\

a_{3,1} & a_{3,2} & a_{3,3} & \dots \\

\vdots & \vdots & \vdots & \ddots

\end{array}

$$

be a doubly infinite array of positive integers, and suppose each

positive integer appears exactly eight times in the array. Prove that

$a_{m,n} > mn$ for some pair of positive integers $(m,n)$.

让

$$

\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\

a_{2,1} & a_{2,2} & a_{2,3} & \点 \\

a_{3,1} & a_{3,2} & a_{3,3} & \点 \\

\vdots & \vdots & \vdots & \ddots

\结束{数组}

$$

是一个双无限正整数数组，并假设每个

正整数在数组中恰好出现八次。证明

$a_{m,n} > mn$ 对于某对正整数 $(m,n)$。